What's the name of a person that is married to more than one person?

Bigamist?

The institution of marrying more than one person is called polygamy. If it is a woman that is married to more than one man, it is called polyandry. If it is a man that is married to more than one woman, it is called polygyny. Bigamy is specifically being married to two people. The person would be a polygamist, polyandrist, polygynist or bigamist; accordingly.